2.1 Review of linear algebra and matrix theory

Linear algebra and matrix theory form the backbone of nonlinear control systems. These mathematical tools help us analyze and manipulate complex systems, transforming abstract concepts into solvable problems.

Matrices, vectors, and operations like decomposition and eigenanalysis are crucial for understanding system behavior. Mastering these concepts allows us to tackle more advanced topics in nonlinear control, setting the stage for deeper insights and practical applications.

Solving linear equations with matrices

Matrix and vector fundamentals

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• Matrices are rectangular arrays of numbers arranged in rows and columns
  • Used to represent linear systems and transformations
  • Example: A 3x3 matrix representing a linear transformation in 3D space
• Vectors are ordered lists of numbers that can be represented as matrices with a single row or column
  • Describe quantities with both magnitude and direction
  • Example: A 3D vector (2, 3, 5) representing a force or displacement
• Matrix addition and subtraction are performed element-wise
  • Example: Adding two 2x2 matrices A and B results in a new 2x2 matrix C, where each element C_ij = A_ij + B_ij
• Matrix multiplication involves multiplying rows by columns and summing the results
  • Example: Multiplying a 2x3 matrix A by a 3x2 matrix B results in a 2x2 matrix C, where each element C_ij = Σ_k (A_ik × B_kj)

Solving linear systems using matrices

• Gaussian elimination transforms the augmented matrix into row echelon form to solve systems of linear equations
  • Involves elementary row operations (row swapping, scaling, and addition) to simplify the matrix
  • Example: Using Gaussian elimination to solve a system of 3 linear equations with 3 unknowns
• Cramer's rule uses determinants to solve systems of linear equations
  • Determinants are scalar values associated with square matrices
  • Example: Applying Cramer's rule to solve a 2x2 system of linear equations
• The inverse of a square matrix A, denoted as A^(-1), is a matrix that, when multiplied by A, yields the identity matrix
  • Used to solve linear equations and analyze linear systems
  • Example: Using the inverse of a 3x3 matrix to solve a system of linear equations Ax = b, where x = A^(-1)b

Matrix decomposition for simplification

LU and QR decomposition

• LU decomposition factors a square matrix into the product of a lower triangular matrix (L) and an upper triangular matrix (U)
  • Simplifies the process of solving linear systems and computing matrix inverses
  • Example: Decomposing a 4x4 matrix A into L and U, where A = LU
• QR decomposition factors a matrix into the product of an orthogonal matrix (Q) and an upper triangular matrix (R)
  • Useful for solving least squares problems and analyzing the stability of linear systems
  • Example: Decomposing a 3x4 matrix A into Q and R, where A = QR

Singular Value Decomposition (SVD) and Cholesky decomposition

• SVD factorizes a matrix into the product of three matrices: U, Σ, and V^T
  • U and V are orthogonal matrices, and Σ is a diagonal matrix containing the singular values
  • Used for data compression, noise reduction, and principal component analysis
  • Example: Applying SVD to a 5x3 matrix A to obtain U, Σ, and V^T
• Cholesky decomposition factors a symmetric, positive-definite matrix into the product of a lower triangular matrix and its transpose
  • More efficient than LU decomposition for solving linear systems and computing matrix inverses
  • Example: Decomposing a 3x3 symmetric, positive-definite matrix A into L, where A = LL^T

Eigenvectors and eigenvalues in linear systems

Eigenvector and eigenvalue fundamentals

• Eigenvectors are non-zero vectors that, when multiplied by a square matrix A, yield a scalar multiple of themselves
  • The scalar multiple is called the eigenvalue
  • Example: Finding the eigenvectors and eigenvalues of a 2x2 matrix A
• Eigenvalues represent the scaling factors by which eigenvectors are stretched or compressed when transformed by a linear system
  • Example: A 2D linear transformation that stretches a vector by a factor of 2 along one eigenvector and compresses it by a factor of 0.5 along another eigenvector

Geometric interpretation and eigenspaces

• Eigenvectors corresponding to distinct eigenvalues are linearly independent and form a basis for the vector space associated with the linear system
  • Example: A 3x3 matrix with three distinct eigenvalues has three linearly independent eigenvectors that form a basis for R^3
• The eigenspace of an eigenvalue is the set of all eigenvectors associated with that eigenvalue, along with the zero vector
  • Example: The eigenspace of an eigenvalue λ = 2 for a 2x2 matrix A consists of all vectors (x, y) that satisfy the equation Av = 2v
• Geometrically, eigenvectors represent the principal directions or axes of a linear transformation, while eigenvalues describe the magnitude of the transformation along those axes
  • Example: A 2D linear transformation that rotates vectors by 45 degrees has eigenvectors along the diagonal lines y = x and y = -x

Stability and behavior of linear systems

Matrix diagonalization and spectral radius

• A matrix is diagonalizable if it has a full set of linearly independent eigenvectors
  • Can be factored as A = PDP^(-1), where D is a diagonal matrix containing the eigenvalues and P is a matrix whose columns are the corresponding eigenvectors
  • Example: Diagonalizing a 3x3 matrix A with three distinct eigenvalues
• The spectral radius of a matrix is the maximum absolute value among its eigenvalues
  • Determines the rate of convergence or divergence of iterative methods and the stability of linear systems
  • Example: A matrix with a spectral radius less than 1 will result in convergent iterations, while a spectral radius greater than 1 will lead to divergence

Matrix properties and stability analysis

• A matrix is positive definite if all its eigenvalues are positive
  • Crucial for ensuring the stability and uniqueness of solutions in optimization problems and control systems
  • Example: A 2x2 matrix A with eigenvalues λ_1 = 2 and λ_2 = 5 is positive definite
• The condition number of a matrix, defined as the ratio of its largest to smallest singular value, measures its sensitivity to perturbations and numerical errors
  • A matrix with a high condition number is considered ill-conditioned and may lead to inaccurate results when solving linear systems or performing matrix operations
  • Example: A 3x3 matrix with singular values σ_1 = 100, σ_2 = 10, and σ_3 = 0.1 has a condition number of 1000, indicating poor conditioning
• The rank of a matrix is the maximum number of linearly independent rows or columns
  • Determines the dimension of the image space and the nullspace of the linear transformation represented by the matrix
  • Example: A 4x3 matrix with rank 2 has a 2-dimensional image space and a 1-dimensional nullspace